Question: Let $h$ be a twice differentiable function, and let $h(8)=5$, $h'(8)=0$, and $h''(8)=-4$. What occurs in the graph of $h$ at the point $(8,5)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(8,5)$ is a minimum point. (Choice B) B $(8,5)$ is a maximum point. (Choice C) C There's not enough information to tell.
Answer: Since $h'(8)=0$, we know that $x=8$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $h$ at this point according to these three cases: If $h''(8)>0$, the graph of $h$ has a minimum point at $x=8$. If $h''(8)<0$, the graph of $h$ has a maximum point at $x=8$. If $h''(8)=0$, the test is inconclusive. [Why is this so?] We are given that $h''(8)=-4<0$. Therefore, $(8,5)$ is a maximum point.